The first term \(\frac{\partial \Omega ^2}{\partial {\bf k}}\)  in the bracket
is a diagonal one corresponding to \(\omega _\sigma \frac{\partial \omega _\sigma }{ \partial {\bf k}}\)  relating the group velocity. The second term in the
bracket \([{\bf \mathcal {A}}, \Omega ^2]\)  gives the off-diagonal elements of
the heat current density, which can be regarded as the contribution from
anomalous velocities similar to the one in the intrinsic anomalous Hall effect.
The Berry connection \({\bf \mathcal {A}}\) , or we can call it Berry vector
potential matrix (the Berry vector potential defined in Ref. [1]},
\({\bf A}^\sigma ({\bf k})\) , is equal to \(i {\bf \mathcal {A}}^{\sigma \sigma }=i\tilde{x}_\sigma ^T \frac{\partial x_\sigma }{\partial {\bf k}}\) ), induces
the anomalous velocities to the heat current, which will take the responsibility
of the PHE. Therefore, the Berry vector potential comes naturally into
the heat current and the PHE. Such a picture is clearer than that in Ref. [1]}.
